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Full text of "Treatise On Analysis Vol-Ii"

18    CANONICAL DECOMPOSITIONS OF A MEASURE        207

the existence, write \JL = / p and v = g - p, where p is a positive measure and
/^ 0, and let M be the p-measurable set of points x e X such thatf(x) > 0, so
that \JL is concentrated on M (13.14.1). Then the measures v' = <pMg - p and
v" = v  v' = (1  <PM)# ' P satisfy the conditions of the theorem. For v" is
disjoint from \JL by virtue of (13.18.1); also if A is a /z-negligible set, then
A n M is p-negligible ((13.14.1) and (13.6,3)), hence A is v'-negligible
(13.14.1) and consequently v' is a measure with base \JL (13.15.5). It is clear that
if v is positive then so are v' and v" ; the formula v' = sup(inf(v, /7/z)) then

n

follows from the corresponding formula (pMg = sup(inf(#, /)), which is

n

valid almost everywhere with respect to p.

A measure /^ on X is said to be diffuse if n({x}) = 0 for all x E X. For
example, Lebesgue measure on R is diffuse. With respect to a diffuse measure
\JL every denumerable set is ^-negligible*, equivalently, a diffuse measure is
concentrated on the complement of any denumerable set. On a denumer-
able discrete space, the only diffuse measure is therefore n = 0. The sum of
two diffuse measures is diffuse (13.16.1). The least upper bound of any set
of diffuse positive measures which is bounded above is diffuse, by (13.15.9)
and (13. 15.4).

(13.18.5)   If \JL is any measure on X, the set A of points xeX such that
\I*\({x}) > 0 -is at most denumerable.

Since X is the union of a sequence of compact sets Kn , it is enough to show
that each of the sets A n Kn is at most denumerable. For this, it is enough to
show that, for each integer m JS> 1 , the set Amn of points x e A n Kn such that
M(M) ^ \l m infinite', now this is immediate, for if B c Amn consists of p
points, then we have p/m \p\(B) ^

A measure /z on X is said to be atomic if it is concentrated on an at most
denumerable set. From this definition and from (1 3.1 8.1 ) it follows immediately
that an atomic measure and a diffuse measure are always disjoint. The sum of
two atomic measures is atomic. The least upper bound of a set of atomic
positive measures which is bounded above is atomic ((13.15.4) and (13.15.9)).

(13.18.6)    Every measure ^ can be expressed uniquely in the form ^ + /x2,
where /^ is a diffuse measure and ju2 an atomic measure.

The uniqueness of the decomposition follows from the fact that an atomic
measure and a diffuse measure are disjoint. To establish the existence of
the decomposition, it is enough to consider the at most denumerable set fr)l fr ^1  e fri* ^i], then we have